reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th8:
  x0 in dom f & f is_continuous_in x0 implies r(#)f is_continuous_in x0
proof
  assume x0 in dom f;
  then
A1: x0 in dom (r(#)f) by VALUED_1:def 5;
  assume
A2: f is_continuous_in x0;
  let s1;
  assume that
A3: rng s1 c= dom(r(#)f) and
A4: s1 is convergent & lim s1=x0;
A5: rng s1 c= dom f by A3,VALUED_1:def 5;
  then
A6: f.x0 = lim (f/*s1) by A2,A4;
A7: f/*s1 is convergent by A2,A4,A5;
  then r(#)(f/*s1) is convergent;
  hence (r(#)f)/*s1 is convergent by A5,RFUNCT_2:9;
  thus (r(#)f).x0 = r*f.x0 by A1,VALUED_1:def 5
    .= lim (r(#)(f/*s1)) by A7,A6,SEQ_2:8
    .= lim ((r(#)f)/*s1) by A5,RFUNCT_2:9;
end;
